79063 If \(\alpha, \beta \neq 0, f(n)=\alpha^{n}+\beta^{n}\) and
\(\left.\left|\begin{array}{ccc} 3 & 1+f(1) & 1+f(2) \\ 1+f(1) & 1+f(2) & 1+f(3) \\ 1+f(2) & 1+f(3) & 1+f(4) \end{array}\right|=\mathbf{k}(1-\alpha)^{2} \right\rvert\,\)
\((1-\beta)^{2}(\alpha-\beta)^{2}\), then \(K\) is equal to

79064 Let \(P\) and \(Q\) be \(3 \times 3\) matrices \(P \neq Q\). If \(P^{3}=Q^{3}\) and \(P^{2} Q=Q^{2} P\), then determinant of \(\left(P^{2}+Q^{2}\right)\) is equal to

79065 If \(A=\left[\begin{array}{ccc}1 & \sin \theta & 1 \\ -\sin \theta & 1 & \sin \theta \\ -1 & -\sin \theta & 1\end{array}\right]\); then for all \(\theta \in\) \(\left(\frac{3 \pi}{4}, \frac{5 \pi}{4}\right)\), det (A) lies in the interval

79066 Let \(A=\left[\begin{array}{ccc}2 & b & 1 \\ b & b^{2}+1 & b \\ 1 & b & 2\end{array}\right]\), where \(b>0\). Then, the minimum value of \(\frac{\operatorname{det}(A)}{b}\) is

79063 If \(\alpha, \beta \neq 0, f(n)=\alpha^{n}+\beta^{n}\) and
\(\left.\left|\begin{array}{ccc} 3 & 1+f(1) & 1+f(2) \\ 1+f(1) & 1+f(2) & 1+f(3) \\ 1+f(2) & 1+f(3) & 1+f(4) \end{array}\right|=\mathbf{k}(1-\alpha)^{2} \right\rvert\,\)
\((1-\beta)^{2}(\alpha-\beta)^{2}\), then \(K\) is equal to

79064 Let \(P\) and \(Q\) be \(3 \times 3\) matrices \(P \neq Q\). If \(P^{3}=Q^{3}\) and \(P^{2} Q=Q^{2} P\), then determinant of \(\left(P^{2}+Q^{2}\right)\) is equal to

79065 If \(A=\left[\begin{array}{ccc}1 & \sin \theta & 1 \\ -\sin \theta & 1 & \sin \theta \\ -1 & -\sin \theta & 1\end{array}\right]\); then for all \(\theta \in\) \(\left(\frac{3 \pi}{4}, \frac{5 \pi}{4}\right)\), det (A) lies in the interval

79066 Let \(A=\left[\begin{array}{ccc}2 & b & 1 \\ b & b^{2}+1 & b \\ 1 & b & 2\end{array}\right]\), where \(b>0\). Then, the minimum value of \(\frac{\operatorname{det}(A)}{b}\) is

79063 If \(\alpha, \beta \neq 0, f(n)=\alpha^{n}+\beta^{n}\) and
\(\left.\left|\begin{array}{ccc} 3 & 1+f(1) & 1+f(2) \\ 1+f(1) & 1+f(2) & 1+f(3) \\ 1+f(2) & 1+f(3) & 1+f(4) \end{array}\right|=\mathbf{k}(1-\alpha)^{2} \right\rvert\,\)
\((1-\beta)^{2}(\alpha-\beta)^{2}\), then \(K\) is equal to

79064 Let \(P\) and \(Q\) be \(3 \times 3\) matrices \(P \neq Q\). If \(P^{3}=Q^{3}\) and \(P^{2} Q=Q^{2} P\), then determinant of \(\left(P^{2}+Q^{2}\right)\) is equal to

79065 If \(A=\left[\begin{array}{ccc}1 & \sin \theta & 1 \\ -\sin \theta & 1 & \sin \theta \\ -1 & -\sin \theta & 1\end{array}\right]\); then for all \(\theta \in\) \(\left(\frac{3 \pi}{4}, \frac{5 \pi}{4}\right)\), det (A) lies in the interval

79066 Let \(A=\left[\begin{array}{ccc}2 & b & 1 \\ b & b^{2}+1 & b \\ 1 & b & 2\end{array}\right]\), where \(b>0\). Then, the minimum value of \(\frac{\operatorname{det}(A)}{b}\) is

79063 If \(\alpha, \beta \neq 0, f(n)=\alpha^{n}+\beta^{n}\) and
\(\left.\left|\begin{array}{ccc} 3 & 1+f(1) & 1+f(2) \\ 1+f(1) & 1+f(2) & 1+f(3) \\ 1+f(2) & 1+f(3) & 1+f(4) \end{array}\right|=\mathbf{k}(1-\alpha)^{2} \right\rvert\,\)
\((1-\beta)^{2}(\alpha-\beta)^{2}\), then \(K\) is equal to

79064 Let \(P\) and \(Q\) be \(3 \times 3\) matrices \(P \neq Q\). If \(P^{3}=Q^{3}\) and \(P^{2} Q=Q^{2} P\), then determinant of \(\left(P^{2}+Q^{2}\right)\) is equal to

79065 If \(A=\left[\begin{array}{ccc}1 & \sin \theta & 1 \\ -\sin \theta & 1 & \sin \theta \\ -1 & -\sin \theta & 1\end{array}\right]\); then for all \(\theta \in\) \(\left(\frac{3 \pi}{4}, \frac{5 \pi}{4}\right)\), det (A) lies in the interval

79066 Let \(A=\left[\begin{array}{ccc}2 & b & 1 \\ b & b^{2}+1 & b \\ 1 & b & 2\end{array}\right]\), where \(b>0\). Then, the minimum value of \(\frac{\operatorname{det}(A)}{b}\) is